https://file.aiquickdraw.com/mathbot/file/1725940525911x9kvcies.jpg

The given limit is:

\[
\lim_{x \to 0} \frac{\sin(5x)}{\sin(10x)}
\]

### Solution:

1. **Direct Substitution**: 
   We cannot directly substitute \( x = 0 \) because both \( \sin(5x) \) and \( \sin(10x) \) will be 0 at \( x = 0 \), leading to the indeterminate form \( \frac{0}{0} \).

2. **Apply Limit Properties**: 
   To resolve this, we can use the standard limit property:
   \[
   \lim_{x \to 0} \frac{\sin(kx)}{kx} = 1
   \]
   for any constant \( k \).

3. **Rewrite the Expression**:
   We manipulate the limit expression by multiplying both the numerator and denominator by \( x \):
   \[
   \lim_{x \to 0} \frac{\sin(5x)}{\sin(10x)} = \lim_{x \to 0} \frac{\frac{\sin(5x)}{x}}{\frac{\sin(10x)}{x}} 
   \]
   Now, we can use the limit property:
   \[
   \lim_{x \to 0} \frac{\sin(5x)}{5x} \times \frac{10x}{\sin(10x)} = \frac{5}{10} = \frac{1}{2}
   \]

Thus, the value of the limit is:
\[
\boxed{\frac{1}{2}}
\]

Would you like further clarification or details?

---

Here are 5 related questions:
1. How would this limit change if the functions involved were \(\sin^2(5x)\) and \(\sin^2(10x)\)?
2. What happens if we change the angle coefficients to different constants, say \( \sin(3x) \) and \( \sin(7x) \)?
3. How do you calculate limits using L'Hopital's Rule in cases of indeterminate forms like \( \frac{0}{0} \)?
4. What are some other common trigonometric limits that are frequently encountered?
5. Can you explain the intuition behind the standard limit property \( \lim_{x \to 0} \frac{\sin(kx)}{kx} = 1 \)?

### Tip:
When dealing with limits involving trigonometric functions, the small-angle approximation \(\sin(x) \approx x\) near zero is incredibly useful for resolving indeterminate forms.

Evaluate the following limit or state that it does not exist: lim (x -> 0) [sin(5x) / sin(10x)]

Learn how to evaluate the limit lim (x -> 0) [sin(5x) / sin(10x)] using standard limit properties for trigonometric functions. This step-by-step solution explains the small-angle approximation and is suitable for high school and undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation